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Consider a circle with unit radius. There are $7$ adjacent sectors, $\text{S}_1, \text{S}_2, \text{S}_3,\dots, \text{S}_7$ in the circle such that their total area is $(1/8)$th of the area of the circle. Further, the area of the $j$-th sector is twice that of the $(j –1)$-th sector, for $j = 2, \dots, 7.$ What the angle, in radians, subtended by the arc of $\text{S}_1$ at the centre of the circle?

  1. $\frac{\pi}{508}$
  2. $\frac{\pi}{2040}$
  3. $\frac{\pi}{1016}$
  4. $\frac{\pi}{1524}$
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